They probably derive moral tenets from a combination of Kantian universalism (the idea, essentially that you should only do things if everyone could do them without adverse reaction), social-benefit utilitarianism (do the most good for the most people), and good old-fashioned playground morality (play nice with others). There are a number of valid morality-based systems that don't require belief in God.
As the Humanist spokesperson says, however, they don't want to be bound by a narrowly defined moral code. Certainly there are troubling implications there. At the very least, a wide interpretation of morality makes it far harder to make societal decisions about right and wrong, opening the possibility that in the confusion, more bad and less good is allowed to occur.
Rrrriiiiiiiight...